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You are asking a good question. As far as I know there are now well established standards that are sufficient for determining the complete viscoelastic response of elastomers. I usually perform the following tests:

Uniaxial tension followed by unloading at at least 3 different strain rates

Uniaxial compression followed by unloading at at least 3 different strain rates

Perhaps also one or more stress relaxation tests

In some cases I then perform additional shear or biaxial tests

I then repeat these experiments at various temperatures if I am interested in determining the temperature response of the material.

I typically use the specimen dimensions that are specified in the ASTM standards for compression and tension experiments of polymers (D0638, D0575).

Yes those standards are valid. Note, however, that I only use the standards to obtain suitable specimen shapes and dimensions. I usually perform the testing slightly differently than what is specified in the standards.

The ASTM standards that I mentioned are not "optimized" for material model calibration. Two problems with the standards are (if I remember right):
1) Only one strain rate is used (better to test at multiple strain rates)
2) Only monotonic loading is used (better to record the response during both the loading and unloading)